Contemporary Mathematics Volume 612, 2014 http://dx.doi.org/10.1090/conm/612/12224
Spectra of Frame Operators with Prescribed Frame Norms
Marcin Bownik and John Jasper
Abstract. We study the set of possible finite spectra of self-adjoint operators with fixed diagonal. In the language of frame theory, this is equivalent to study of the set of finite spectra of frame operators with prescribed frame norms. We show several properties of such sets. We also give some numerical examples illustrating our results.
1. Introduction The concept of frames in Hilbert spaces was originally introduced in the context of nonharmonic Fourier series by Duffin and Schaeffer [12] in 1950’s. The advent of wavelet theory brought a renewed interest in frame theory as is attested by now classical books of Daubechies [11], Meyer [26], and Mallat [24]. For an introduction to frame theory we refer to the book by Christensen [10].
Definition 1.1. A sequence {fi}i∈I in a Hilbert space H is called a frame if there exist 0 2010 Mathematics Subject Classification. Primary: 42C15, 47B15, Secondary: 46C05. Key words and phrases. Diagonals of self-adjoint operators, the Schur-Horn theorem, the Pythagorean theorem, the Carpenter theorem, spectral theory. This work was partially supported by a grant from the Simons Foundation (#244422 to Marcin Bownik). The second author was supported by NSF ATD 1042701. c 2014 American Mathematical Society 65 This is a free offprint provided to the author by the publisher. Copyright restrictions may apply. 66 MARCIN BOWNIK AND JOHN JASPER his collaborators [7–9] characterized norms of finite tight frames in terms of “fun- damental frame inequality” using frame potential methods and gave an explicit and algorithmic construction of finite tight frames with prescribed norms. Kor- nelson and Larson [22] studied a similar problem for infinite dimensional Hilbert spaces using projection decomposition. Antezana, Massey, Ruiz, and Stojanoff [1] established the connection of this problem with the infinite dimensional Schur-Horn problem and gave refined necessary conditions and sufficient conditions. A beau- tifully simple and complete characterization of Parseval frame norms was given by Kadison [18, 19], which easily extends to tight frames by scaling. The authors [4] have extended this result to the non-tight setting to characterize frame norms with prescribed optimal frame bounds. The second author [17] has characterized diagonals of self-adjoint operators with three points in the spectrum. This yields a characterization of frame norms whose frame operator has two point spectrum. Finally, the authors [5,6] have recently extended this result to operators with finite spectrum. The above mentioned research was aimed primarily at characterizing diago- nals of operators (or frame norms) with prescribed spectrum (or frame operator). However, it is equally interesting to consider a converse problem of characterizing spectra of operators with prescribed diagonal. In the language of frames, we are asking for possible spectra of frame operators for which the sequence of frame norms {||fi||}i∈I is prescribed. That is, given n ∈ N and a sequence {di}i∈I in [0, 1] we consider the set n (1.2) An = An({di})= (A1,...,An) ∈ (0, 1) : ∀j= k Aj = Ak 2 ∃ frame {f } ∈ such that ∀ ∈ d = ||f || and its i i I i I i i frame operator S satisfies σ(S)={A1,...,An, 1} . In this work we shall always assume that there exists α ∈ (0, 1) such that di + (1 − di) < ∞. di<α di≥α Otherwise, it can be shown by Theorems 2.3 and 2.7 that An({di})isthesetofall points in (0, 1)n with distinct coordinates. The second author [17, Theorem 7.1] has shown that the set A1 is always finite (possibly empty). In this paper we show some further properties such as a characterization of sequences {di} for which A1 is nonempty. We also prove that the set A2 consists of a countable union of line segments. Moreover, one endpoint of each of these line segments must lie in the boundary of the unit square. Finally, we show that the sets An are nonempty for all n ≥ 2 under the assumption that infinitely many di’s satisfy di ∈ (0, 1). Moreover, we prove the optimality of this result. 2. Background results about Schur-Horn type theorems The classical Schur-Horn theorem [16,28] characterizes diagonals of self-adjoint (Hermitian) matrices with given eigenvalues. It can be stated as follows, where HN N N is N dimensional Hilbert space over R or C, i.e., HN = R or C . Theorem . { }N { }N 2.1 (Schur-Horn theorem) Let λi i=1 and di i=1 be real sequences in nonincreasing order. There exists a self-adjoint operator E : HN →HN with This is a free offprint provided to the author by the publisher. Copyright restrictions may apply. SPECTRA OF FRAME OPERATORS WITH PRESCRIBED FRAME NORMS 67 eigenvalues {λi} and diagonal {di} if and only if N N n n (2.1) di = λi and di ≤ λi for all 1 ≤ n ≤ N. i=1 i=1 i=1 i=1 The necessity of (2.1) is due to Schur [28] and the sufficiency of (2.1) is due to Horn [16]. It should be noted that (2.1) can equivalently be stated in terms of the convexity condition (2.2) (d1,...,dN ) ∈ conv{(λσ(1),...,λσ(N)):σ ∈ SN }. This characterization has attracted a significant interest and has been generalized in many remarkable ways. Some major milestones are the Kostant convexity theo- rem [23] and the convexity of moment mappings in symplectic geometry [3,14,15]. Moreover, the problem of extending Theorem 2.1 to an infinite dimensional di- mensional Hilbert space H was also investigated. Neumann [27] gave an infinite dimensional version of the Schur-Horn theorem phrased in terms of ∞-closure of the convexity condition (2.2). Neumann’s result can be considered an initial, al- beit somewhat crude, solution of this problem. The first fully satisfactory progress was achieved by Kadison. In his influential work [18, 19] Kadison discovered a characterization of diagonals of orthogonal projections acting on H. Theorem 2.2 (Kadison). Let {di}i∈I be a sequence in [0, 1] and α ∈ (0, 1). Define C(α)= di,D(α)= (1 − di). di<α di≥α 2 There exists an orthogonal projection on (I) with diagonal {di}i∈I if and only if either: (i) C(α)=∞ or D(α)=∞,or (ii) C(α) < ∞ and D(α) < ∞,and (2.3) C(α) − D(α) ∈ Z. The work by Gohberg and Markus [13] and Arveson and Kadison [2] extended the Schur-Horn Theorem 2.1 to positive trace class operators. This has been further extended to compact positive operators by Kaftal and Weiss [21]. These results are stated in terms of majorization inequalities as in (2.1), see also [20] for a detailed survey of recent progress on infinite Schur-Horn majorization theorems and their connections to operator ideals. Antezana, Massey, Ruiz, and Stojanoff [1] refined the results of Neumann [27]. Moreover, they showed the following connection between Schur-Horn type theorems and the existence of frames with prescribed norms and frame operators, see [1, Proposition 4.5] and [4, Proposition 2.3]. Theorem 2.3. Let {di}i∈I be a bounded sequence of positive numbers. Let S be a positive self-adjoint operator on a Hilbert space H. Then the following are equivalent: (i) there exists a frame {fi}i∈I in H with the frame operator S such that 2 di = ||fi|| for all i ∈ I, (ii) there exists a larger Hilbert space K⊃Hand a self-adjoint operator E acting on 2(I), which is unitarily equivalent with S ⊕ 0,where0 is the zero operator acting on KH, such that its diagonal Eei,ei = di for all i ∈ I. This is a free offprint provided to the author by the publisher. Copyright restrictions may apply. 68 MARCIN BOWNIK AND JOHN JASPER The authors [4] have recently shown a variant of the Schur-Horn theorem for a class of locally invertible self-adjoint operators on H. Theorem 2.4. Let 0 di (2.5) ∃n ∈ N0 nA ≤ C ≤ A + B(n − 1) + D. As a corollary of Theorems 2.3 and 2.4 we obtain the characterization of se- quences of frame norms. Corollary 2.5. Let 0 ∃N ∈ N ∃k ∈ Z C − D = NA+ kB and C ≥ (N + k)A. Finally, the authors [5] showed the following characterization of diagonals of self-adjoint operators with finite spectrum. Theorem 2.7 becomes a foundation on which all subsequent results in this paper will be derived. We will often use this result under a convenient normalization that B =1. Theorem . { }n+1 2.7 Let Aj j=0 be an increasing sequence of real numbers such ∈ N { } that A0 =0and An+1 = B, n .Let di i∈I be a sequence in [0,B] with d = (B − d )=∞.Foreachα ∈ (0,B), define i i (2.6) C(α)= di and D(α)= (B − di). di<α di≥α Then, there exists a self-adjoint operator E with diagonal {di}i∈I and σ(E)= {A0,A1,...,An+1} if and only if either: (i) C(B/2) = ∞ or D(B/2) = ∞,or (ii) C(B/2) < ∞ and D(B/2) < ∞, (and thus C(α),D(α) < ∞ for all α ∈ (0,B)),andthereexistN1,...,Nn ∈ N and k ∈ Z such that: n (2.7) C(B/2) − D(B/2) = AjNj + kB, j=1 This is a free offprint provided to the author by the publisher. Copyright restrictions may apply. SPECTRA OF FRAME OPERATORS WITH PRESCRIBED FRAME NORMS 69 and for all r =1,...,n, r n (2.8) (B − Ar)C(Ar)+ArD(Ar) ≥ (B − Ar) AjNj + Ar (B − Aj )Nj . j=1 j=r+1 3. Spectral set In light of Theorem 2.3 we shall adopt the following definition for the set (1.2). Definition 3.1. Suppose that {di}i∈I is a sequence in [0, 1]. For a given n ∈ N we define the spectral set n A ({d })= (A ,...,A ) ∈ (0, 1) : ∀ A = A , ∃ self-adjoint operator n i 1 n j=k j k 2 E on (I) with σ(E)={0,A1,...,An, 1} and diagonal {di} . − ∞ In order to apply Theorem 2.7 we shall assume that di = (1 di)= . This is not a true limitation. Indeed, the case when di < ∞ or (1 − di) < ∞ requires an application of a finite rank analogue of Theorem 2.7. This leads effectively to a finite dimensional case where the analysis is actually simpler, but also less interesting; see Remark 3.3. Furthermore, we will consider only sequences in the set F := {di} :0≤ di ≤ 1and∃ α ∈ (0, 1) such that di + (1 − di) < ∞ . di<α di≥α n Otherwise, Theorem 2.7(i) implies that An({di}) is the set of all points in (0, 1) with distinct coordinates, which is not interesting. 3.1. Three point spectrum. In this subsection we will look at properties of A { } A { } the set 1( di ). In [17]itwasshownthat 1( di ) is a finite (possibly empty) set for all {di}∈F. Recall that we assume di = (1 − di)=∞ so that we can apply Theorem 2.7. Definition 3.2. For a sequence {di}i∈I ∈F set η = C(1/2) − D(1/2) −C(1/2) − D(1/2), and define the function m :(0, 1) → Z by m(A)=C(A) − D(A) − η. Remark 3.1. Note that for A By Theorem 2.7 we have A ∈A1({di}) if and only if there exists N ∈ N and k ∈ Z such that (3.2) C(A) − D(A)=AN + k and (3.3) (1 − A)C(A)+AD(A) ≥ (1 − A)AN. This is a free offprint provided to the author by the publisher. Copyright restrictions may apply. 70 MARCIN BOWNIK AND JOHN JASPER In principle, to verify that A ∈A1 one would need to check the inequality (3.3) for all N and k such that (3.2) holds. The next theorem shows that one needs only to check that (3.3) holds for a particular N ∈ N. Theorem 3.3. Let {di}i∈I ∈F,andletη be as in Definition 3.2.FixA ∈ (0, 1) and set q =min{j ∈ N: jA − η ∈ Z}. The number A is in A1({di}) if and only if (3.4) (1 − A)C(A)+AD(A) ≥ (1 − A)Aq. Remark 3.2. If there is no j ∈ N so that jA − η ∈ Z,thenwehaveq = min ∅ = ∞. In this case the inequality (3.4) does not hold and we conclude that A/∈A1. Proof. Assume that A ∈A1. By Theorem 2.7 there exists N ∈ N and k ∈ Z such that (3.2) and (3.3) hold. From (3.2) we see that NA − η ∈ Z.Thus,q ≤ N and (3.4) holds. Conversely, assume that (3.4) holds. By Theorem 2.7 it is enough to show that (3.2) and (3.3) hold with N = q and k = m(A)−qA+η. It is clear that (3.3) holds, since this is exactly (3.4). Finally, we have C(A) − D(A)=m(A)+η = qA + k, which is exactly (3.2). Theorem 3.3 gives a finite algorithm to check whether a given number A ∈ (0, 1) A is in 1.Foreach (1 − A)C(A)+AD(A) 1 ≤ j ≤ (1 − A)A we compute the quantity jA − η. Then, by Theorem 3.3 A ∈A1 if and only if one of these numbers is an integer. Theorem 3.4. Let {di}i∈I ∈F,andletη be as in Definition 3.2.Ifη =0 , then η ∈A1({di}).Inparticular,if{di} is not the diagonal of a projection, then A1({di}) = ∅. Proof. By Theorem 3.3 it is enough to show that (3.4) holds with A = η. Note that q =min{j ∈ N: jη − η ∈ Z} =1. Thus, it is enough to verify that (3.5) (1 − η)C(η)+ηD(η) ≥ (1 − η)η. If m(η) ≥ 0, then C(η) ≥ C(η) − D(η)=m(η)+η ≥ η. If m(η) ≤−1, then D(η) ≥ D(η) − C(η)=−m(η) − η ≥ 1 − η. In either case we conclude that (3.5) holds. The next theorem characterizes the sequences {di} such that A1({di})=∅. This is a free offprint provided to the author by the publisher. Copyright restrictions may apply. SPECTRA OF FRAME OPERATORS WITH PRESCRIBED FRAME NORMS 71 Theorem 3.5. Let {di}i∈I ∈F,andletη be as in Definition 3.2.Theset A1 = ∅ if and only if η =0and (3.6) C(1/2) + D(1/2) < 1. That is, A1 is empty if and only if {di} is the diagonal of a projection and 1/2 ∈A/ 1. Proof. We begin by assuming that A1 = ∅. By Theorem 3.4 we must have η =0,otherwiseη ∈A1 and thus A1 = ∅. Thus, we may assume η =0.Since 1/2 ∈A/ 1, Theorem 3.3 shows (3.6). Conversely, assume that η = 0 and (3.6) holds. Since C(1/2) − D(1/2) ∈ Z we conclude that C(1/2) = D(1/2) < 1/2andm(1/2) = 0. For the sake of contradiction assume that there is some A ∈A1 ∩ (0, 1/2]. From (3.2) we see that A ∈ Q.Thatis,A = p/q for some p, q ∈ N with gcd(p, q) = 1. From (3.2) we have (3.7) −m(A)=m(1/2) − m(A)=|{i : A ≤ di < 1/2}| Using (3.7) and the definition of m(A)wehave (1 − A)C(A)+AD(A)=−Am(A)+C(A)=−Am(A)+ di di A≤di<1/2 ≤−Am(A)+C(1/2) − A|{i : A ≤ di < 1/2}| = C(1/2) 1 < ≤ 1 − A ≤ (1 − A)p =(1− A)Aq. 2 This implies that (3.4) does not hold, and by Theorem 3.3 A/∈A1.Consequently, A1 ∩ (0, 1/2] = ∅. A similar argument shows that A1 ∩ (1/2, 1) = ∅, and thus A1 = ∅. Remark 3.3. Recall that in this section it is assumed that di = (1−di)= ∞ so that Theorem 2.7 applies. However, all of the results in this subsection hold under the weaker assumption that both di ≥ 1and (1 − di) ≥ 1. In the case that {di} or {1 − di} is summable the proofs are similar to those above. The difference being that the applications of Theorem 2.7 are replaced by a finite rank version of the Schur-Horn theorem [21]. 3.2. Finite point spectrum. In this subsection we will look at properties of the set An({di}), where n ≥ 2. Definition 3.6. Let {di}i∈I ∈Fand let η be as in Definition 3.2. By Theorem 2.7 a point (A1,...,An)with0 This is a free offprint provided to the author by the publisher. Copyright restrictions may apply. 72 MARCIN BOWNIK AND JOHN JASPER n Given (N1,...,Nn) ∈ N define the set (3.10) AN1,...,Nn { } n ( di )= (A1,...,An):0 As an immediate consequence of Theorem 2.7 we have that A { } ◦AN1,...,Nn { } (3.11) n( di )= σ n ( di ), n (N1,...,Nn)∈N σ∈Σn n where ◦ denotes the action of the permutation group Σn on R . Theorem 3.7. Let {di}i∈I ∈F be such that (3.12) {i ∈ I : di ∈ (0, 1)} is infinite, AN1,...,Nn { } and let η be as in Definition 3.2.Then,theset n ( di ) is nonempty if one of the following holds: (i) the sequence {di} is a diagonal of a projection, i.e., η =0,andbothof the sets {i ∈ I : di ∈ (0, 1/2)} and {i ∈ I : di ∈ (1/2, 1)} are infinite, (ii) η>0, N1 =1,and{i ∈ I : di ∈ (1/2, 1)} is infinite, (iii) η>0, Nn =1,and{i ∈ I : di ∈ (0, 1/2)} is infinite, Proof. By our hypothesis (3.12) and C(1/2),D(1/2) < ∞,wehavethat{i : di ∈ (0,ε)} or {i : di ∈ (1 − ε, 1)} are infinite for every ε>0. This implies that at least one of the following holds: (3.13) ∀α∈(0,1) C(α) > 0 and lim D(α)=∞, α→0+ (3.14) ∀α∈(0,1) D(α) > 0 and lim C(α)=∞. α→1− First we shall prove that (i) or (ii) imply the existence of 0 <ε<1 − η such that for any A1 ∈ (η, η + ε)andA2 ∈ (1 − ε, 1), there exist A3 < ... < An such that (3.9) holds. Case (i). Suppose that {di} is a diagonal of a projection, i.e., η =0,andboth (3.13) and (3.14) hold. Thus, there exists ε>0 such that n (3.15) D(A1) ≥ Nj for all A1 ∈ (0,ε), j=1 (3.16) C(A2) ≥ N1 + N2 for all A2 ∈ (1 − ε, 1). Once A1 and A2 satisfying (3.15) and (3.16) are chosen we will show inductively that there exist A3 <... Indeed, once A1,...,Ar−1, r ≥ 3, are defined, by (3.14) we can choose Ar suffi- ciently close to 1 such that r n C(Ar) ≥ Nj and (1 − Ar) Nj ≤ D(Ar−1). j=1 j=r This is a free offprint provided to the author by the publisher. Copyright restrictions may apply. SPECTRA OF FRAME OPERATORS WITH PRESCRIBED FRAME NORMS 73 Since n n (1 − Aj)Nj ≤ (1 − Ar) Nj j=r j=r this inductive process yields (A1,...,An) such that (3.15) and (3.17) are satisfied. Thus, (3.9) holds. Case (ii). We assume that η>0, N1 = 1, and (3.14) holds. There are two subcases to consider. Suppose that C(η) >η. By (3.14) this implies that there exists 1 − η>ε>0 such that (3.16) holds and n (3.18) C(A1) ≥ A1 + ε Nj for all A1 ∈ (η, η + ε). j=2 Hence, for any A1 ∈ (η, η + ε)andA2 ∈ (1 − ε, 1) we have n n (1 − A1)C(A1)+A1D(A1) ≥ (1 − A1) A1 +(1− A2) Nj ≥ A1 (1 − Aj)Nj . j=2 j=1 Suppose that C(η) ≤ η.SinceC(α) − D(α) ≡ η mod 1 for all α, by (3.14) we have D(η) > 1 − η. Then again by (3.14) we can choose 1 − η>ε>0 such that (3.16) holds and n (3.19) D(A1) ≥ 1 − A1 + ε Nj for A1 ∈ (η, η + ε). j=2 Hence, for any A1 ∈ (η, η + ε)andA2 ∈ (1 − ε, 1) we have n n (1 − A1)C(A1)+A1D(A1) ≥ A1 1 − A1 +(1− A2) Nj ≥ A1 (1 − Aj)Nj . j=2 j=1 In either case, by an inductive argument as in case (i) one can show that there exist A3 <... for some 0 Theorem 3.8. Let {di}∈F.ThesetAn({di}) is nonempty for each n ≥ 2 if and only if {i: di ∈ (0, 1)} is infinite. Proof. Assume that n ≥ 2 and (3.12) holds. Theorem 3.7 and the identity (3.11) shows that An = ∅ unless we are in the special case when {di} is a diagonal of a projection, η = 0, and only one of the sets {i ∈ I : di ∈ (0, 1/2)} or {i ∈ I : di ∈ (1/2, 1)} is infinite. Without loss of generality we can assume that {i ∈ I : d ∈ (1/2, 1)} is finite since the other case is done by symmetry. This implies that i k0 = di < ∞, where I0 = {i ∈ I : di ∈ [0, 1)}. i∈I0 This is a free offprint provided to the author by the publisher. Copyright restrictions may apply. 74 MARCIN BOWNIK AND JOHN JASPER Moreover di = ∞ implies that I1 = {i ∈ I : di =1} is infinite. Using the finite rank Schur-Horn theorem [4, Theorem 3.2] one can show that { } there exists a self-adjoint operator E0 with diagonal di i∈I0 and spectrum σ(E0)= {0,A1,...,An} for some 0 In order to study more subtle properties of the set An({di})itisusefultoprove the following lemma. Lemma 3.9. Let {di}i∈I ∈Fand let η be as in Definition 3.2. Then the function f :(0, 1) → (0, ∞) defined by f(α)=(1− α)C(α)+αD(α) is piecewise linear, continuous, concave, and it satisfies (3.20) lim f(α) = lim f(α)=0. α→0+ α→1− Moreover, f (α) ≡−η (mod 1) for every α ∈ (0, 1) \{di : i ∈ I}. Proof. The continuity of f at each α ∈ (0, 1) \{di : i ∈ I} is clear from the definition. For α = di we see that 0 lim f(α)=(1− d ) d + d (1 − d )=f(d ), − i0 i i0 i i0 α→d i0 ≥ di This is a free offprint provided to the author by the publisher. Copyright restrictions may apply. SPECTRA OF FRAME OPERATORS WITH PRESCRIBED FRAME NORMS 75 −η (mod 1) for α ∈ (0, 1)\{di : i ∈ I}. We deduce that f is linear on any subinterval I ⊂ (0, 1), which does not contain any of di’s. Moreover, f (α) is decreasing on the set where it is defined. Consequently, f is concave. Finally, to prove (3.20) observe that for all 0 <α<β<1 (1 − β)C(β)=(1− β)C(α)+(1− β) di α≤di<β ≤ (1 − β)C(α)+(1− β)β|{i: α ≤ d <β}| i ≤ (1 − β)C(α)+β (1 − di)=(1− β)C(α)+βD(α) − βD(β). α≤di<β Rearranging gives (3.21) f(β)=(1− β)C(β)+βD(β) ≤ (1 − β)C(α)+βD(α). It can similarly be shown that (3.21) holds for 0 <β<α<1. From this we deduce that for any α ∈ (0, 1) lim sup f(β) ≤ C(α) and lim sup f(β) ≤ D(α). β→0+ β→1− Since C(α) → 0asα → 0+, D(α) → 0asα → 1−,andf(β) ≥ 0 for all β ∈ (0, 1) we have lim f(β) = lim f(β)=0. β→0+ β→1− Remark 3.4. Suppose that 0 = A0 N1 N2 Nn Let g(α)=(1− α)C (α)+αD (α), where C (α)andD (α) are defined by (2.6) for { }σ the sequence dj j=1. Then, the majorization inequalities (2.8) can be restated as (3.22) f(α) ≥ g(α)forα = Ar,r=1,...,n. Since f is a concave function and g is a piecewise linear function with knots at A1,...,An, (3.22) is equivalent to (3.23) f(α) ≥ g(α) for all α ∈ (0, 1). Thus, the majorization inequality (2.8) is equivalent to an inequality (3.23) between { } { }σ two functions defined as above out of two sequences di i∈I and dj j=1,resp. This observation will play a key role in the proof of the following lemma. Lemma 3.10. Let {di}i∈I ∈Fbe a sequence in [0, 1].LetNj ∈ N, j = ∈AN1,...,Nn { } 1,...,n. Suppose that (A1,...,An) n ( di ). Then, for any choice of r, s =1,...,n, r This is a free offprint provided to the author by the publisher. Copyright restrictions may apply. 76 MARCIN BOWNIK AND JOHN JASPER ∈AN1,...,Nn { } we have (A1,...,An) n ( di ). Proof. Let h be the function defined as in Lemma 3.9 corresponding to the sequence ( A1,...,A 1 ,A 2,...,A 2 ,...,A n,...,A n ). N1 N2 Nn { }n By (3.26) we have that (3.8) holds for (Aj,Nj ) j=1. Moreover, in order to establish { }n ≥ (3.9) for (Aj,Nj ) j=1, by Remark 3.4 it suffices to show that g(α) h(α) for all α ∈ (0, 1). This follows immediately from the following identity ⎧ ⎪0 α ∈ (0,A ) ∪ (A , 1), ⎨⎪ r s N (A − α) α ∈ (A ,A ), (3.27) h(α) − g(α)= r r r r ⎪N (A − A )=N (A − A ) α ∈ (A ,A ), ⎩⎪ r r r s s s r s − ∈ Ns(α As) α (As,As). ∈ The proof of (3.27) is a calculation which we present only for α (Ar,Ar). By comparing common terms appearing in g and h we see by (3.24) and (3.25) that they cancel out except when j = r, s. That is, by (3.26) we have − − − − − − − h(α) g(α)=(1 α)NrAr + αNs(1 As) αNr(1 Ar) αNs(1 As) − − − = NrAr + αNs αAr αNs = Nr(Ar α). This shows that (3.9) holds for (A1,...,An) and completes the proof of Lemma 3.10. As a consequence of Lemma 3.10 and some standard results in majorization theory [25] we have the following result which bears a close resemblance to the Schur-Horn Theorem 2.1. Theorem 3.11. Let {di}i∈I ∈Fbe a sequence in [0, 1].LetNj ∈ N, j = ∈AN1,...,Nn { } 1,...,n. Suppose that (A1,...,An) n ( di ). Then, for any 0 Proof. For simplicity we shall sketch the proof only in the special case N1 = ... = Nn = 1, where results from the majorization theory are readily applicable. By [25, Lemma 2.B.1] sequence (A1,...,An) can be derived from (A1,...,An)by a successive finite application of T -transforms, also known as convex moves. These operations correspond to successive applications of Lemma 3.10. Also by analyzing the proof of [25, Lemma 2.B.1] it is apparent that these T -transforms preserve strict monotonicity of (A1,...,An)ateachstep. 3.3. Four point spectrum. As a consequence of Lemma 3.10 we have the AN1,N2 { } following result about 2 ( di ). Lemma 3.12. Let {di}i∈I ∈F be a sequence in [0, 1] and N1,N2 ∈ N.Then, AN1,N2 { } the set 2 ( di ) consists of a finite number (possibly zero) of line segments with slopes −N1/N2. One endpoint of each of these line segments must lie in the boundary of the unit square (0, 1)2. This is a free offprint provided to the author by the publisher. Copyright restrictions may apply. SPECTRA OF FRAME OPERATORS WITH PRESCRIBED FRAME NORMS 77 Proof. ∈AN1,N2 { } ∈ Z By (2.7) any (A1,A2) 2 ( di ) must satisfy for some k C(1/2) − D(1/2) = N1A1 + N2A2 + k. The above equation defines a line with slope −N1/N2 in (A1,A2) plane which can intersect the unit square (0, 1)2 only for finitely many values of k ∈ Z. By Lemma 3.10 for any point (A1,A2)with0 Corollary 3.13. Let {di}i∈I ∈Fbe a sequence in [0, 1] satisfying (3.12). Then, the set A2({di}) is nonempty and it consists of a countable union of line segments. Moreover, one endpoint of each of these line segments must lie in the boundary of the unit square. We shall illustrate Corollary 3.13 for the symmetric geometric sequence already studied in [17]. 1.0 1.0 0.8 0.8 0.6 0.6 0.4 0.4 0.2 0.2 0.0 0.0 0.0 0.2 0.4 0.6 0.8 1.0 0.0 0.2 0.4 0.6 0.8 1.0 1.0 1.0 0.8 0.8 0.6 0.6 0.4 0.4 0.2 0.2 0.0 0.0 0.0 0.2 0.4 0.6 0.8 1.0 0.0 0.2 0.4 0.6 0.8 1.0 Figure 1. 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